When \(M\)  is a smooth oriented manifold, the Euler class of the tangent bundle has a beautiful geometric interpretation (cf. Proposition 12.8 in [1]}): for any section \(s\)  of \(\xi \)  which intersects the zero section transversely, the Euler class is the Poincare dual to the homology class given by the zero locus of \(s\)  (once appropriate orientation is given to this zero locus). This comes out particularly simply when the dimension of the base space is also \(r\) , since then the zero locus is a discrete set of points. For each such point \(p\) , transversality of \(s\)  and the zero section \(z\)  tells us that \(T_{s(p)}(E)=s_*(T_p(M))\oplus z_*(T_p(M))\) . Define \(c(p)\)  to be 1 if the orientations of the two sides of this equation agree and \(-1\)  if they disagree. Then the Poincare dual of \(e(E)\)  is
\(e(E)\frown [M]=\sum _pc(p)\) 
